cardinal functions and integral functions, by mircea e. selariu, florentin smarandache and marian...

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INTERNATIONAL JOURNAL OF GEOMETRY

Vol. 1 (2012), No. 1, 27 - 40

CARDINAL FUNCTIONS AND

INTEGRAL FUNCTIONS

MIRCEA E. SELARIU, FLORENTIN SMARANDACHE andMARIAN NITU

Abstract. This paper presents the correspondences of the eccentricmathematics of cardinal and integral functions and centric mathematics,or ordinary mathematics. Centric functions will also be presented in theintroductory section, because they are, although widely used in undulatoryphysics, little known.

In centric mathematics, cardinal sine and cosine are de…ned as well as theintegrals. Both circular and hyperbolic ones. In eccentric mathematics, allthese central functions multiplies from one to in…nity, due to the in…nity of possible choices where to place a point. This point is called eccenter S (s; ")

which lies in the plane of unit circle UC(O; R = 1) or of the equilateral unityhyperbola HU(O; a = 1; b = 1). Additionally, in eccentric mathematics thereare series of other important special functions, as aex, bex, dex, rex,etc. If we divide them by the argument , they can also become cardinaleccentric circular functions, whose primitives automatically become integraleccentric circular functions.

All supermatematics eccentric circular functions (SFM-EC) can be of vari-able excentric , which are continuous functions in linear numerical eccen-tricity domain s 2 [1; 1], or of centric variable , which are continuous forany value of s. This means that s 2 [1; +1].

————————————–Keywords and phrases: C-Circular , CC- C centric, CE- C Eccentric,

CEL-C Elevated, CEX-C Exotic, F-Function, FMC-F Centric Mathemat-ics, M- Matemathics, MC-M Centric, ME-M Excentric, S-Super, SM- SMatematics, FSM-F Supermatematics FSM-CE- FSM Eccentric Circulars,FSM-CEL- FSM-C Elevated, FSM-CEC- FSM-CE- Cardinals, FSM-CELC-FSM-CEL Cardinals

(2010) Mathematics Subject Classi…cation: 32A17



28 M. Selariu, F . Sma randache and M. N itu

1. INTRODUCTION: CENTRIC CARDINAL SINE FUNCTION

According to any standard dictionary, the word "cardinal" is synonymouswith "principal", "essential", "fundamental".

In centric mathematics (CM), or ordinary mathematics, cardinal is, onthe one hand, a number equal to a number of …nite aggregate, called thepower of the aggregate, and on the other hand, known as the sine cardinalsinc(x) or cosine cardinal cosc(x), is a special function de…ned by the centriccircular function (CCF). sin(x) and cos(x) are commonly used in undulatoryphysics (see Figure 1) and whose graph, the graph of cardinal sine, which iscalled as "Mexican hat" (sombrero) because of its shape (see Figure 2).

Note that sinc(x) cardinal sine function is given in the speciality litera-ture, in three variants

sinc(x) =

(1; for x = 0

sin(x)

x; for x 2 [1; +1] n 0

(1)

=sin(x)

x= 1 x2

6+

x4

120 x76

5040+

x8

362880+ 0[x]11

=+1Xn=0

(1)nx2n

(2n + 1)!! sinc(

2) =

2

;

d(sinc(x))

dx=

cos(x)

x sin(x)

x2= cosc(x) sinc(x)

x;

(2) sinc(x) =sin(x)

x

(3) sinca(x) =sin(xa )

xa

It is a special function because its primitive, called sine integral and denotedSi(x)

Centric circular cardinalsine functions

Modi…ed centric circularcardinal sine functions

Figure 1: The graphs of centric circular functions cardinal sine, in 2D, as

known in literature



Cardinal functions and integral functions 29

Figure 2: Cardinal sine function in 3D Mexican hat (sombrero)

Si(x) = Z x

0

sin(t)

tdt = Z

x

0sinc(t) dt(4)

= x x3

18+

x5

600

x7

35280+

x9

3265920+ 0[x]11

= x x8

3:3!+

x5

5:5! x7

7:7!+ : : :

=+1Xn=0

(1)nx2n

(2n + 1)2(2n)!; 8x 2 R

can not be expressed exactly by elementary functions, but only by expansionof power series, as shown in equation (4). Therefore, its derivative is

(5) 8x 2 R; Si0

(x) =d(Si(x))

dx =sin(x)

x = sinc(x);

an integral sine function Si(x), that satis…es the di¤erential equation

(6) x f 000

(x) + 2f 00

(x) + x f 0

(x) = 0 ! f (x) = Si(x):

The Gibbs phenomenon appears at the approximation of the square witha continuous and di¤erentiable Fourier series (Figure 3 right I). This op-eration could be substitute with the circular eccentric supermathematicsfunctions (CE-SMF), because the eccentric derivative function of eccentricvariable can express exactly this rectangular function (Figure 3 N top) orsquare (Figure 3 H below) as shown on their graphs (Figure 3 J left).

1 cos x=2p 1sin(x=2)2

;

fx;; 2:01g

12 4xPSi nc[2(2k1)x];

fk; ngfn; 5gfx; 0; 1g




12dex[(;

2 ); S (1; 0)]

Gibbs phenomenon fora square wave with n = 5

and n = 10

Figure 3: Comparison between the square function, eccentric derivative

and its approximation by Fourier serial expansion

Integral sine function (4) can be approximated with su¢cient accuracy.The maximum di¤erence is less than 1%, except the area near the origin.By the CE-SMF eccentric amplitude of eccentric variable

(7) F () = 1:57 aex[; S (0:6; 0)];

as shown on the graph on Figure 5.

Sin R x;

fx;

20; 20

g

Sin

R (x + Iy)

fx;

20; 20

g;

fy;

3; 3

g

Figure 4: The graph of integral sine function Si(x) N compared with thegraph CE-SMF Eccentric amplitude 1; 57aex[; S (0; 6; 0)] of eccentric

variable H




Figure 5: The di¤erence between integral sine and CE-SMF eccentricamplitude F () = 1; 5aex[; S (0; 6;0] of eccentric variable

2. ECCENTRIC CIRCULAR SUPERMATHEMATICS CARDINALFUNCTIONS, CARDINAL ECCENTRIC SINE (ECC-SMF)

Like all other supermathematics functions (SMF),they may be eccentric(ECC-SMF), elevated (ELC-SMF) and exotic (CEX-SMF), of eccentric vari-able , of centric variable 1;2 of main determination, of index 1, or secondarydetermination of index 2. At the passage from centric circular domain tothe eccentric one, by positioning of the eccenter S (s; ") in any point in the

plane of the unit circle, all supermathematics functions multiply from one toin…nity. It means that in CM there exists each unique function for a certaintype. In EM there are in…nitely many such functions, and for s = 0 one willget the centric function. In other words, any supermathematics functioncontains both the eccentric and the centric ones.

Notations sexc(x) and respectively, Sexc(x) are not standard in the lit-erature and thus will be de…ned in three variants by the relations:

(8) sexc(x) =sex(x)

x=

sex[; S (s; s)]

of eccentric variable and

(8’) Sexc(x) =Sex

(x

)x =Sex

[; S

(s; s

)]

of eccentric variable .

(9) sexc(x) =sex(x)

x;

of eccentric variable , noted also by sexc(x) and

(9’) Sexc(x) =sex(x)

x=

Sex[; S (s; s)]

;

of eccentric variable , noted also by Sexc(x)

(10) sexca(x) =sexx

a

xa

=sex

;




of eccentric variable , with the graphs from Figure 6 and Figure 7.

(10’) Sexa(x) =

Sex xa

xa

=

Sex aa

aa

Sin ArcSin [s Sin ()]

fs; 0; +1g; f;; 4gSin

ArcSin [s Sin ()]

fs;1; 0g; f;; g

Sin ArcSin [s Sin ()]

fs; 0; 1g; f;; gFigure 6: The ECCC-SMF graphs sexc1[; S (s; ")] of eccentric variable

Sin +ArcSin [s Sin ()]

fs; 0; 1g; f;4; 4gSin

ArcSin [0.1s Sin ()]

fs;10; 0g; f;; g

Sin ArcSin [0.1s Sin ()]

fs; 0:1; 0g; f;; gFigure 7: Graphs ECCC-SMF sexc2[; S (s; ")]; eccentric variable




3. ECCENTRIC CIRCULAR SUPERMATHEMATICSFUNCTIONS CARDINAL ELEVATED SINE AND COSINE

(ECC- SMF-CEL)

Supermathematical elevated circular functions (ELC-SMF), elevated sinesel() and elevated cosine cel(), is the projection of the fazor/vector

~r = rex() rad() = rex[; S (s; ")] rad()

on the two coordinate axis X S and Y S respectively, with the origin in the

eccenter S (s; "), the axis parallel with the axis x and y which originate inO(0; 0).

If the eccentric cosine and sine are the coordinates of the point W (x; y),by the origin O(0; 0) of the intersection of the straight line d = d + [ dae\,revolving around the point S (s; "), the elevated cosine and sine are the samecoordinates to the eccenter S (s; "); ie, considering the origin of the coordi-nate straight rectangular axes XSY /as landmark in S (s; "). Therefore, therelations between these functions are as follows:

(11)

x = cex() = X + s cos(") = cel() + s cos(")y = Y + s sin(") = sex() = sel() + s sin(")

Thus, for " = 0, ie S eccenter S located on the axis x > 0;sex() = sel(),and for " =

2 ;cex() = cel(), as shown on Figure 8.On Figure 8 were represented simultaneously the elevated cel() and the

sel() graphics functions, but also graphs of cex() functions, respectively,for comparison and revealing sex() elevation Eccentricity of the functionsis the same, of s = 0:4, with the attached drawing and sel() are " =

2 , andcel() has " = 0.

Figure 8: Comparison between elevated supermathematics function and

eccentric functions




Figure 9: Elevated supermathematics function and cardinal eccentricfunctions celc(x) J and selc(x) I of s = 0:4

Figure 10: Cardinal eccentric elevated supermathematics functioncelc(x) J and selc(x) I

Elevate functions (11) divided by become cosine functions and cardinalelevated sine, denoted celc() = [; S ] and selc() = [; S ], given by theequations

(12)

8><>:

X = celc() = celc[; S (s; ")] = cexc() s cos(s)

Y = selc() = selc[; S (s; ")] = sexc() s sin(s)

with the graphs on Figure 9 and Figure 10.




4. NEW SUPERMATHEMATICS CARDINAL ECCENTRICCIRCULAR FUNCTIONS (ECCC-SMF)

The functions that will be introduced in this section are unknown in math-ematics literature. These functions are centrics and cardinal functions orintegrals. They are supermathematics eccentric functions amplitude, beta,radial, eccentric derivative of eccentric variable [1], [2], [3], [4], [6], [7] cardi-nals and cardinal cvadrilobe functions [5].

Eccentric amplitude function cardinal aex(), denoted as

(x) = aex[; S (s; ")]; x ;

is expressed in

(13) aexc() = aex(

= aex[; S (s; s)]

= arcsin[s sin( s]

and the graphs from Figure 11.

Sin () ; fs; 0; 1g; f;4; +4g

Figure 11: The graph of cardinal eccentric circular supermathematicsfunction aexc()

The beta cardinal eccentric function will be

(14) bexc() =bex()

=

bex[S (s; s)]

=

arcsin[s sin( s)]

;

with the graphs from Figure 12.

Figure 12: The graph of cardinal eccentric circular supermathematicsfunction bexc() (

fs;

1; 1

g;

f;

4; 4

g)




The cardinal eccentric function of eccentric variable is expressed by

(15) rexc1;2() =rex()

=rex[; S (s; s)]

=s cos( s)

p 1 s2sin( s)

and the graphs from Figure 13, and the same function, but of centric variable is expressed by

(16) Rexc(1;2) =Rex(1;2)

1;2

=Rex[1;2S (s; s)]

1;2

=

p 1 + s2 2s cos(1;2 s)

1;2

sCos()+p 1[sSin()]2 ;

fs; 0; 1g; f;4; 4gsCos()p 1[s Sin()]2

;

fs;1; 0g; f;4; 4gsCos()p 1[sSin()]2

;

fs; 0; 1g; f;4; 4gFigure 13: The graph of cardinal eccentric circular supermathematical

function rexc1;2()

And the graphs for Rexc(1), from Figure 14.

Figure 14: The graph of cardinal eccentric radial circularsupermathematics function Rex c()

An eccentric circular supermathematics function with large applications,representing the function of transmitting speeds and/or the turning speeds of all known planar mechanisms is the derived eccentric dex1;2() and Dex(1;2),

functions that by dividing/reporting with arguments and, respectively,




lead to corresponding cardinal functions, denoted dexc1;2(), respectivelyDexc(1;2) and expressions

(17) dexc1;2() =dex1;2()

=

dex1;2[; S (s; s)]

=

1 scos(")1s2sin2(")

(18)

Dexc(1;2) =Dex(1;2)

1;2=

Dexf[1;2S (s; s)]g1;2

=

p 1 + s2 2s cos(1;2 s)

1;2

the graphs on Figure 15.

Figure 15: The graph of supermathematical cardinal eccentric radialcircular function dex c1()

Because Dex(1;2) = 1dex1;2()

results that Dexc(1;2) = 1dexc1;2()

siq ()

and coq () are also cvadrilobe functions, dividing by their arguments leadto cardinal cvadrilobe functions siqc() and coqc() obtaining with the ex-pressions

(19) coqc() =coq ()

=

coq [S (s; s)]

=

cos( s)

p

1 s2sin2( s)

(20) siqc() =siq ()

=

siq [S (s; s)]

=

sin( s)

p

1 s2cos2( s)

the graphs on Figure 16.

Figure 16: The graph of supermathematics cardinal cvadrilobe functionceqc() J and siqc() I




It is known that, by de…nite integrating of cardinal centric and eccentricfunctions in the …eld of supermathematics, we obtain the corresponding

integral functions.Such integral supermathematics functions are presented below. For zero

eccentricity, they degenerate into the centric integral functions. Otherwisethey belong to the new eccentric mathematics.

5. ECCENTRIC SINE INTEGRAL FUNCTIONS

Are obtained by integrating eccentric cardinal sine functions (13) and are

(21) sie(x) = Z x

0

sexc()

d

with the graphs on Figure 17 for the ones with the eccentric variable x .

Figure 17: The graph of eccentric integral sine function sie 1(x)N andsie 2(x)H

Unlike the corresponding centric functions, which is denoted Sie(x), theeccentric integral sine of eccentric variable was noted sie(x), without thecapital S , which will be assigned according to the convention only for theECCC-SMF of centric variable. The eccentric integral sine function of cen-tric variable, noted Sie(x) is obtained by integrating the cardinal eccentricsine of the eccentric circular supermathematics function, with centric vari-able

(22) Sexc(x) = Sexc[; S (s; ")];




thus

(23) Sie(x) =Z x0

Sex[; S (s; "]

;

with the graphs from Figure 18.

Figure 18: The graph of eccentric integral sine function sie 2(x)

6. C O N C L U S I O N

The paper highlighted the possibility of inde…nite multiplication of car-

dinal and integral functions from the centric mathematics domain in theeccentric mathematics’s or of supermathematics’s which is a reunion of thetwo mathematics. Supermathematics, cardinal and integral functions werealso introduced with correspondences in centric mathematics, a series newcardinal functions that have no corresponding centric mathematics.

The applications of the new supermathematics cardinal and eccentricfunctions certainly will not leave themselves too much expected.

References

[1] Selariu, M. E.,Eccentric circular functions

, Com. I C onferinta National¼a de Vibratiiîn Constructia de Masini, Timisoara , 1978, 101-108.[2] Selariu, M. E., Eccentric circular functions and their extension , Bul .St. Tehn. al I.P.T.,

Seria Mecanic¼a, 25(1980), 189-196.

[3] Selariu, M. E., Supermathematica , C om.VII Conf. International¼a. De Ing. Manag. siTehn.,TEHNO’95 Timisoara, 9(1995), 41-64.

[4] Selariu, M. E., Eccentric circular supermathematic functions of centric variable,

Com.VII Conf. International¼a. De Ing. Manag. si Tehn.,TEHNO’98 Timisoara, 531-548.

[5] Selariu, M. E., Quadrilobic vibration systems , The 11th International Conference on

Vibration Engineering, Timisoara, Sept. 27-30, 2005, 77-82.[6] Selariu, M. E., Supermathematica. Fundaments, Vol.II , Ed. Politehnica, Timisoara,

2007.[7] Selariu, M. E., Supermathematica. Fundaments, Vol.II , Ed. Politehnica, Timisoara,

2011 (forthcoming).




Received: December, 2011

POLYTEHNIC UNIVERSITY OF TIMISOARA, ROMANIAE-mail address : [email protected]

UNIVERSITY OF NEW MEXICO-GALLUP, USAE-mail address : [email protected]

INSTITUTUL NATIONAL DE CERCETARE - DEZVOLTAREPENTRU ELECTROCHIMIE SI MATERIE CONDENSAT ¼ATIMISOARA, ROMANIAE-mail address : [email protected]

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